Generalized <i>q</i>-Difference Equations for <i>q</i>-Hypergeometric Polynomials with Double <i>q</i>-Binomial Coefficients

Abstract

In this paper, we apply a general family of basic (or q-) polynomials with double q-binomial coefficients as well as some homogeneous q-operators in order to construct several q-difference equations involving seven variables. We derive the Rogers type and the extended Rogers type formulas as well as the Srivastava-Agarwal-type bilinear generating functions for the general q-polynomials, which generalize the generating functions for the Cigler polynomials. We also derive a class of mixed generating functions by means of the aforementioned q-difference equations. The various results, which we have derived in this paper, are new and sufficiently general in character. Moreover, the generating functions presented here are potentially applicable not only in the study of the general q-polynomials, which they have generated, but indeed also in finding solutions of the associated q-difference equations. Finally, we remark that it will be a rather trivial and inconsequential exercise to produce the so-called (p,q)-variations of the q-results, which we have investigated here, because the additional forced-in parameter p is obviously redundant.

Keywords

• homogeneous <i>q</i>-difference operator
• double <i>q</i>-binomial coefficients
• <i>q</i>-difference equations
• <i>q</i>-hypergeometric polynomials
• generating functions